reserve Z for open Subset of REAL;

theorem Th20:
  for n be Nat, r,x be Real st r > 0 holds Maclaurin(
sin, ].-r,r.[,x).(2*n) = 0 & Maclaurin(sin, ].-r,r.[,x).(2*n+1) = (-1) |^ n * x
|^ (2*n+1) / ((2*n+1)!) & Maclaurin(cos, ].-r,r.[,x).(2*n) = (-1) |^ n * x |^ (
  2*n) / ((2*n)!) & Maclaurin(cos, ].-r,r.[,x).(2*n+1) = 0
proof
A1: |.0-0.|=0 by ABSVALUE:2;
  let n be Nat, r,x be Real;
  assume r > 0;
  then
A2: 0 in ].0-r,0+r.[ by A1,RCOMP_1:1;
  then
A3: 0 in dom(cos | ].-r,r.[) by Th17;
A4: dom(((-1) |^ n) (#) (cos | ].-r,r.[)) = dom(cos | ].-r,r.[) by
VALUED_1:def 5
    .= ].-r,r.[ by Th17;
A5: Maclaurin(sin, ].-r,r.[,x).(2*n+1) = (diff(sin,].-r,r.[).(2*n+1)).0 * (
  x-0) |^ (2*n+1) / ((2*n+1)!) by TAYLOR_1:def 7
    .= ((-1) |^ n (#) cos | ].-r,r.[).0 * x |^ (2*n+1) / ((2*n+1)!) by Th19
    .= (((-1) |^ n) * (cos | ].-r,r.[).0) * x |^ (2*n+1) / ((2*n+1)!) by A2,A4,
VALUED_1:def 5
    .= ((-1) |^ n) * cos.0 * x |^ (2*n+1) / ((2*n+1)!) by A3,FUNCT_1:47
    .= (-1) |^ n * x |^ (2*n+1) / ((2*n+1)!) by SIN_COS:30;
A6: dom(((-1) |^ n) (#) (sin | ].-r,r.[)) = dom(sin | ].-r,r.[) by
VALUED_1:def 5
    .= ].-r,r.[ by Th17;
A7: 0 in dom(sin | ].-r,r.[) by A2,Th17;
A8: Maclaurin(cos, ].-r,r.[,x).(2*n) = (diff(cos,].-r,r.[).(2*n)).0 * (x-0)
  |^ (2*n) / ((2*n)!) by TAYLOR_1:def 7
    .= ((-1) |^ n (#) cos | ].-r,r.[).0 * x |^ (2*n) / ((2*n)!) by Th19
    .= (((-1) |^ n) * (cos | ].-r,r.[).0) * x |^ (2*n) / ((2*n)!) by A2,A4,
VALUED_1:def 5
    .= ((-1) |^ n) * cos.0 * x |^ (2*n) / ((2*n)!) by A3,FUNCT_1:47
    .= (-1) |^ n * x |^ (2*n) / ((2*n)!) by SIN_COS:30;
A9: dom(((-1) |^ (n+1)) (#) (sin | ].-r,r.[)) = dom(sin | ].-r,r.[) by
VALUED_1:def 5
    .= ].-r,r.[ by Th17;
A10: Maclaurin(cos, ].-r,r.[,x).(2*n+1) = (diff(cos,].-r,r.[).(2*n+1)).0 * (
  x-0) |^ (2*n+1) / ((2*n+1)!) by TAYLOR_1:def 7
    .= ((-1) |^ (n+1) (#) sin | ].-r,r.[).0 * x |^ (2*n+1) / ((2*n+1)!) by Th19
    .= (((-1) |^ (n+1)) * (sin | ].-r,r.[).0) * x |^ (2*n+1) / ((2*n+1)!) by A2
,A9,VALUED_1:def 5
    .= ((-1) |^ (n+1)) * sin.0 * x |^ (2*n+1) / ((2*n+1)!) by A7,FUNCT_1:47
    .= 0 by SIN_COS:30;
  Maclaurin(sin, ].-r,r.[,x).(2*n) = (diff(sin,].-r,r.[).(2*n)).0 * (x-0)
  |^ (2*n) / ((2*n)!) by TAYLOR_1:def 7
    .= ((-1) |^ n (#) sin | ].-r,r.[).0 * x |^ (2*n) / ((2*n)!) by Th19
    .= (((-1) |^ n) * (sin | ].-r,r.[).0) * x |^ (2*n) / ((2*n)!) by A2,A6,
VALUED_1:def 5
    .= ((-1) |^ n) * sin.0 * x |^ (2*n) / ((2*n)!) by A7,FUNCT_1:47
    .= 0 by SIN_COS:30;
  hence thesis by A5,A8,A10;
end;
